每个准完全数至少有八个不同的素因子
Every quasiperfect number has at least eight distinct prime factors
AI总结：
本文通过三个中学代数引理消除了准完全数下界的阻碍，结合机器计算首次完整证明准完全数若存在则至少有8个不同素因子，填补了此前研究的覆盖缺口。
AI中文摘要：
目前还未发现准完全数（满足σ(n)=2n+1的数），其不同素因子的个数存在下界；自1982年以来，Hagis和Cohen给出的下界ω≥7一直成立，该下界的阻碍源于一类“深叶”，纯枚举法无法对其终止（中间素数的扫描边界达到8×10^8，且指数维度无界）。本文用三个中学代数层面的引理消除了该阻碍，分别是判别式准则、二次剩余筛和多线性求解器，依次消除最后一个素数q、中间素数p和指数维度，将非终止搜索转化为有限判定。在此基础上，所有“3整除n且ω=7”的381个分支及其79751212个深叶被消除，账目完全闭合且无任何解；互补情形“3不整除n且ω=7”收缩为单个分支，被直接消除，因此证明不依赖任何无法独立复现验证的已发表定理。结合ω≤6的机器消除（定理B4），本文得到主定理：任何准完全数（若存在）满足ω(n)≥8，这是自1982年Hagis和Cohen以来该下界的首次进展。完整计算已通过三种算法架构（CPU和GPU）的七个独立闭合账目复现，均无任何解且账目完全闭合，引理层在Lean中形式化（259个定理，零“sorry”）。2023年Zemann的预印本通过不同计算得到了相同下界，但本文对其公开代码的审计发现存在35个可行指数的覆盖缺口，因此据本文所知，该消除是首个完整证明。代码、账目和Lean资源可从作者处获取。
英文摘要：
No quasiperfect number ($σ(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $ω\ge 7$ of Hagis--Cohen has stood since 1982, obstructed by a family of ``deep leaves'' on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra --- a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver --- which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of ``$3 \mid n$ and $ω= 7$'' and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case ``$3 \nmid n$ and $ω= 7$'' collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $ω\le 6$ (Theorem B4), this yields the main theorem: \emph{any quasiperfect number, if one exists, satisfies $ω(n) \ge 8$} --- the first advance of this bound since Hagis--Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero \texttt{sorry}). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.
